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Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2n + 1) 2 = 4n(n + 1) + 1, and n(n + 1) is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number. The difference between any two odd perfect squares is a multiple of 8.
The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane.In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. .